Alfie Pavey’s first-half goal was enough to send Dover into the second round of the FA Trophy.
The striker, playing against his former side, outmuscled Ryan Woodford and rolled the ball past home stopper Ben Dudzinski.
Andy Hessenthaler’s men will now host Harrogate town in the next round on Saturday 12th January.
Earlier, Ibra Sekajja found himself one on one with Lee Worgan, but slipped at the crucial moment, allowing the visiting keeper the close down the striker’s shot.
Wes Tarbuck then volleyed Moussa Diarra’s defensive header back towards goal with interest, but saw his effort bounce narrowly wide.
It was Whites who went ahead on 19 minutes, though Havant had chances to level.
Wes Fogden dragged a shot wide and Sekajja went even closer from an angle, but Worgan safely watched the ball fly past his far post.
Prior to the half-time whistle, Kadell Daniel’s thunderous strike whizzed by Dudzinski’s side netting.
Following the restart, the linesman flag denied Dover a second goal.
Dudzinski let Daniel’s curler slip through his fingers and though Pavey tapped in, his goal was ruled out for offside.
Worgan then came to Whites’ rescue, diving low to hold onto Sekajja’s low drive.
At the other end Jamie Allen run through, but Dudzinski beat the attacker to the ball.
The Hawks stopper faced a similar situation with substitute Effiong bearing down on goal, but the keeper was out quickly once more to clear the danger just in time.
And the centre forward had two more opportunities to confirm Dover’s spot in the second round.
After Dudzinski dropped a corner, Effiong saw his volley blocked on the line by Jordan Rose and then after sprinting clear, placed his shot just off target.
The away side looked to seal the win but Anthony Jeffrey sent a 20-yard shot wide and Effiong struck the foot of the post from a similar distance.
Earlier, boss Andy Hessenthaler rotated his squad, handing starts to Diarra, Jeffrey, Worgan, Daniel and Joe Bedford.
|Date||Time||League||Season||Full Time||Attendance||Man of the Match|
|8th January 2019||7:45 pm||The FA Trophy||2018-2019||90'||229||Alfie Pavey|